European Commission says United Kingdom registrants will not be able to renew their domains after Brexit is completed.

It’s looking more and more likely that they will lose their domain names because they will no longer be eligible registrants. Michele Neylon of Blacknight noted that the European Commission sent a letter yesterday explaining how this will work:

As of the withdrawal date, undertakings and organisations that are established in the United Kingdom but not in the EU and natural persons who reside in the United Kingdom will no longer be eligible to register .eu domain names or, if they are .eu registrants, to renew .eu domain names registered before the withdrawal date.

Exactly when these domain names will be taken away isn’t clear, but the Commission suggests it could happen on the date of Brexit itself, currently scheduled for March 30th, 2019. This won’t have a catastrophic impact (the .eu TLD never really took off in the UK) but will still affect the 317,000 or .eu domains estimated to be registered in the UK.
You may want to consider migrating to a new domain now, so you can set up your redirects and new marketing push a year in advance.
Of course, there are lots of proxies people can use to generally get around residency and other eligibility requirements.